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Register a new userConcise lectures on selected topics of von Neumann algebras
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Fumio Hiai
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A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970s by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.
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A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.
We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general linear groups induces a bijection of their idempotents with the following additional property: If two idempotents or their two complements generate the same right ideal, then so does their image. This generalises work on regular rings, such include von Neumann factors of type $I_{n}$, $n < infty$.
The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type ${I!I}_1$. In the absence of a trace, one can still define the index of a conditional expectation associated to a subfactor and look for expectations that minimize the index. This value is called the minimal index of the subfactor. We report on our analysis, contained in [GL19], of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily finite, nor factorial) with finite-dimensional centers. Our results generalize some aspects of the Jones index for multi-matrix inclusions (finite direct sums of matrix algebras), e.g., the minimal index always equals the squared norm of a matrix, that we call emph{matrix dimension}, as it is the case for multi-matrices with respect to the Bratteli inclusion matrix. We also mention how the theory of minimal index can be formulated in the purely algebraic context of rigid 2-$C^*$-categories.
The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $tau$-measurable operators (affiliated with a semi-finite von Neumann algebra equipped with a trace $tau$), (ii) positive elements in Haagerups $L^p$-spaces, (iii) semi-finite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of non-commutative (Hilsum) $L^p$-spaces.
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
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